Tessellations, lattices and polynomials in lens spaces (in Spanish)

Abstract

Lens spaces form a classical family of 3-manifolds whose classification can be stated as an arithmetic condition. Starting from a Heegaard diagram one can associate to them a graph embedded in the torus and, lifting it to the plane, obtain a periodic lattice and a tessellation that make part of that arithmetic visible. In this talk I explain how the Bollobás–Riordan polynomial of the graph leads to extremal problems about paths, Steiner trees and minimal supports in the lattice. I present structural results that translate coefficients of the polynomial into geometric information, together with the conjecture that this polynomial completely distinguishes lens spaces. Much of this work relied on artificial intelligence tools: to turn abstract constructions into pictures, to write and debug experiments, and to stress-test claims that at first seemed obvious. I discuss what these tools actually contributed, which errors they helped detect, and why mathematical verification remains indispensable.